Fractional Calculus: Models And Numerical Methods (Second Edition)

by Juan J Trujillo, Enrico Scalas, Kai Diethelm, Dumitru Baleanu

Overview

This book will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods; moreover, we will introduce some applied topics, in particular fractional variational methods which are used in physics, engineering or economics. We will also discuss the relationship between semi-Markov continuous-time random walks and the space-time fractional diffusion equation, which generalizes the usual theory relating random walks to the diffusion equation. These methods can be applied in finance, to model tick-by-tick (log)-price fluctuations, in insurance theory, to study ruin, as well as in macroeconomics as prototypical growth models.All these topics are complementary to what is dealt with in existing books on fractional calculus and its applications currently existing in the market. This book will be written with a trade-off in mind between full mathematical rigor and the needs of readers coming from different applied areas of science and engineering. In particular, the numerical methods listed in the book are presented in a readily accessible way that immediately allows the readers to implement them on a computer in a programming language of their choice.The second edition of the book has been expanded and now includes a discussion of additional, newly developed numerical methods for fractional calculus and a chapter on the application of fractional calculus for modeling processes in the life sciences.

Product Details

ISBN-13:	9789813140035
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/17/2016
Series:	Series On Complexity, Nonlinearity And Chaos , #5
Pages:	476
Product dimensions:	6.10(w) x 9.10(h) x 1.20(d)

Table of Contents

Preface vii

Preface to the Second Edition xxi

1 Preliminaries 1

1.1 Fourier and Laplace Transforms 2

1.2 Special Functions and Their Properties 4

1.2.1 The Gamma function and related special functions 5

1.2.2 Hypergeometric functions 7

1.2.3 Mittag-Leffler functions 8

1.3 Fractional Operators 9

1.3.1 Riemann-Liouville fractional integrals and fractional derivatives 10

1.3.2 Caputo fractional derivatives 15

1.3.3 Liouville fractional integrals and fractional derivatives. Marchaud derivatives 18

1.3.4 Generalized exponential functions 23

1.3.5 Hadamard type fractional integrals and fractional derivatives 28

1.3.6 Fractional integrals and fractional derivatives of a function with respect to another function 33

1.3.7 Grünwald-Letnikov fractional derivatives 36

1.3.8 Variable-order fractional derivatives 38

2 A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations 39

2.1 The Approximation of Fractional Differential and Integral Operators 40

2.1.1 Methods based on quadrature theory 41

2.1.2 Grünwald-Letnikov methods 45

2.1.3 Lubich's fractional linear multistep methods… 47

2.2 Direct Methods for Fractional Ordinary Differential Equations 52

2.2.1 The basic idea 52

2.2.2 Quadrature-based direct methods 53

2.3 Indirect Methods for Fractional Ordinary Differential Equations 55

2.3.1 The basic idea 55

2.3.2 An Adams-type predictor-corrector method 57

2.3.3 Deng's modification of the fractional Adams-Bashforth-Moulton method 61

2.3.4 The Cao-Burrage-Abdullah approach 63

2.4 Linear Multistep Methods 65

2.5 Spectral Methods 68

2.6 Other Methods 72

2.6.1 A decomposition technique 72

2.6.2 The variational iteration method 74

2.6.3 Methods based on nonclassical representations of fractional differential operators 75

2.6.4 Collocation 76

2.7 Methods for Terminal Value Problems 77

2.8 Numerical Methods for Multi-Term Fractional Differential Equations and Multi-Order Fractional Differential Systems 80

2.9 The Extension to Fractional Partial Differential Equations 86

2.9.1 General formulation of the problem 86

2.9.2 Examples 90

3 Specific Efficient Methods for tire Solution of Ordinary and Partial Fractional Differential Equations 95

3.1 Methods for Ordinary Differential Equations 95

3.1.1 Dealing with non-locality: The finite memory principle, nested meshes, and the approaches of Deng and Li 95

3.1.2 Parallelization of algorithms 101

3.1.3 When and when not to use fractional linear multistep formulas 105

3.1.4 The use of series expansions 108

3.1.5 The generalized Adams methods as an efficient tool for multi-order fractional differential equations 110

3.1.6 Two classes of singular equations as application examples 120

3.2 Methods for Partial Differential Equations 126

3.2.1 The method of lines 126

3.2.2 Backward difference formulas for time-fractional parabolic and hyperbolic equations 129

3.2.3 Other methods 138

3.2.4 Methods for equations with space-fractional operators 140

4 Generalized Stirling Numbers and Applications 143

4.1 Introduction 143

4.2 Stirling Functions of the First Kind s(α, k) with Complex First Argument α 146

4.2.1 Equivalent definitions 147

4.2.2 Multiple sum representations. The Riemann Zeta function 152

4.3 General Stirling Functions of the First Kind s(α, β) with Complex Arguments 154

4.3.1 Definition and main result 154

4.3.2 Differentiability of the s(α, β); The zeta function encore 164

4.3.3 Recurrence relations for s(α, β) 166

4.4 Stirling Functions of the Second Kind S(α, k) with Complex First Argument α 168

4.4.1 Stirling functions S(α, k), α ≥ 0, and their representations by Liouville and Marchaud fractional derivatives 169

4.4.2 Stirling functions S(α, k), α < 0, and then-representations by Liouville fractional integrals 172

4.4.3 Stirling functions S(α, k), α ∈ C, and their representations 173

4.4.4 Stirling functions S(α, k), α ∈ C, and recurrence relations 175

4.4.5 Further properties and first applications of Stirling functions S(α, k), α ∈ C 178

4.4.6 Applications of Stirling functions S(α, k) (α ∈ C) to Hadamard-type fractional operatos 184

4.5 Generalized Stirling Functions of the Second Kind S(n, β) with Complex Second Argument β 189

4.5.1 Definition and some basic properties 190

4.5.2 Main properties 197

4.6 Generalized Stirling Functions of the Second Kind S(α, β) with Complex Arguments 202

4.0.1 Basic properties 203

4.6.2 Representations by Liouville fractional operators 209

4.6.3 First application 212

4.6.4 Special examples 216

4.7 Connections Between the Generalized Stirling Functions of the First and the Second Kind Applications 219

4.7.1 Coincidence relations 220

4.7.2 Results from sampling analysis 221

4.7.3 Generalized orthogonality properties 223

4.7.4 The s(α, k) connecting two types of fractional derivatives 224

4.7.5 The representation of a general fractional difference operator via s(α, k) 228

5 Fractional Variational Principles 233

5.1 Fractional Eider-Lagrange Equations 235

5.1.1 Introduction and survey of results 235

5.1.2 Fractional Euler-Lagrange equations for discrete and continuous systems 238

5.1.3 Fractional Lagrangian formulation of field systems 240

5.1.4 Fractional Euler-Lagrange equations with delay 241

5.1.5 Fractional discrete Euler-Lagrange equations 247

5.1.6 Fractional Lagrange-Finsler geometry 248

5.1.7 Applications 251

5.2 Fractional Hamiltonian Dynamics 259

5.2.1 Introduction and overview of results 259

5.2.2 Fractional Hamiltonian analysis for discrete and continuous systems 260

5.2.3 Fractional Hamiltonian formulation for constrained systems 262

5.2.4 Applications 266

5.3 Fractional Variational Problems Within Shifted Orthonormal Legendre Polynomials 270

5.4 Fractional Euler-Lagrange Equations Model for Freely Oscillating Dynamical Systems 284

5.5 Bateman-Feshbach-Tikochinsky Oscillator With Fractional Derivative 289

6 Continuous-Time Random Walks and Fractional Diffusion Models 291

6.1 Introduction 291

6.2 The Definition of Continuous-Time Random Walks 293

6.3 Fractional Diffusion and Limit Theorems 316

7 Applications of Continuous-Time Random Walks 321

7.1 Introduction to Financial Markets 321

7.2 Models of Price Fluctuations in Financial Markets 323

7.3 Simulation 326

7.4 Option Pricing 328

7.5 Other Applications 338

7.5.1 Application to insurance and economics 338

7.5.2 Applications to physics 340

8 Selected Applications of Fractional Calculus Models in the Life Sciences 345

8.1 The Spreading of an Epidemic 345

8.1.1 Basic facts about dengue fever and the classical mathematical model 346

8.1.2 The fractional version of the model 349

8.2 Biological Reactive Systems 352

8.2.1 A mathematical model for the bio-ethanol fermentation process 353

8.2.2 A concrete example and its key properties 355

8.3 The Evolution of the Concentration of a Drug in the Human Body 365

8.3.1 The classical compartmental model of pharmacokinetics 365

8.3.2 Derivation of a fractional order model 367

8.3.3 Further geticralizations 372

Appendix A Source Codes 375

A.1 The Adams-Bashforth-Moulton Method 375

A.2 Lubich's Fractional Backward Differentiation Formulas 387

A.3 Time-fractional Diffusion Equations 397

A.4 Computation of the Mittag-Leffler Function 400

A.5 Monte Carlo Simulation of CTRW 401

Bibliography 405

Index 445